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Mirrors > Home > MPE Home > Th. List > sspnv | Structured version Visualization version GIF version |
Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspnv.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspnv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | eqid 2610 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
3 | eqid 2610 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2610 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
5 | eqid 2610 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
6 | eqid 2610 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
7 | sspnv.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 26963 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) |
9 | 8 | simprbda 651 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 NrmCVeccnv 26823 +𝑣 cpv 26824 ·𝑠OLD cns 26826 normCVcnmcv 26829 SubSpcss 26960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-sm 26836 df-nmcv 26839 df-ssp 26961 |
This theorem is referenced by: sspg 26967 ssps 26969 sspmlem 26971 sspmval 26972 sspz 26974 sspn 26975 sspimsval 26977 sspph 27094 bnsscmcl 27108 minvecolem2 27115 hhshsslem1 27508 hhshsslem2 27509 |
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